Lattice basis reduction is an important field in Number Theory. In general, a lattice basis is said to be of good quality when the basis vectors are relatively short and close to being orthogonal.
We cite three famous types of reduction to get a good basis:
Minkowski-reduced basis, Korkin-Zolotarev-reduced (or Hermite-reduced) basis, and \(LLL\) -reduced basis
for Lenstra-Lenstra-Lovász [1]}[2]}. A basis is said to be \(LLL\) -reduced if it has been processed by the \(LLL\)  algorithm. This algorithm, given an input basis of a lattice, outputs a new basis in polynomial time where the new basis respects some criteria, see e.g. [2]}. The \(LLL\) -reduction is widely used in practice to improve the quality of a basis.
The basis \(\mathcal {B}_1\)  in Figure REF  is Minkowski-, KZ-, and Voronoi-reduced.
